find f'(x) if f(x)=5e^x^2
find f'(x) if f(x)=a^x
f = 5e^(x^2))
use chain rule:
f = 5e^u
f' = 5e^u u'
f' = 5e^(x^2) * 2x = 10x e^(x^2)
f = a^x
recall that a = e^(ln a)
f = e^(lna * x)
f' = lna * e^(lna * x) = a^x lna
Just something you may find interesting:
y = u^v
lny = v lnu
1/y y' = lnu v' + v/u u'
y' = u^v lnu v' + vu^(v-1) u'
Note that
if v is a constant n, v' = 0 and y' = nu^(n-1) u' is just the power rule
if u is a constant a, u' = 0 and y' = a^v lna v' as in your problem.
cool, eh?
so a^x would be undefined?
no. a is just a number, like e.
Is e^x undefined?
What prompted that question?
To find the derivative of a function, we can use the power rule of differentiation.
1. For f(x) = 5e^(x^2):
To find f'(x), we need to apply the chain rule. The chain rule states that if we have a composite function, f(g(x)), the derivative is given by f'(g(x)) * g'(x).
In this case, f(x) = 5e^(x^2), so we can write it as f(x) = 5e^u, where u = x^2. The derivative of e^u is simply e^u, and the derivative of u = x^2 is 2x. Applying the chain rule, we get:
f'(x) = 5e^(x^2) * 2x = 10xe^(x^2).
Therefore, the derivative of f(x) = 5e^(x^2) is f'(x) = 10xe^(x^2).
2. For f(x) = a^x:
To differentiate a function of the form f(x) = a^x, we apply the exponential function rule. The derivative of a^x is given by the natural logarithm of the base a multiplied by a^x.
Therefore, the derivative of f(x) = a^x is f'(x) = ln(a) * a^x.
Please note that in the second equation, a is a constant, and ln(a) represents the natural logarithm of a.